![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfcnv | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfcnv.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfcnv | ⊢ Ⅎ𝑥◡𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnv 5527 | . 2 ⊢ ◡𝐴 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐴𝑦} | |
2 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑥𝑧 | |
3 | nfcnv.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
5 | 2, 3, 4 | nfbr 5077 | . . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦 |
6 | 5 | nfopab 5098 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ 𝑧𝐴𝑦} |
7 | 1, 6 | nfcxfr 2953 | 1 ⊢ Ⅎ𝑥◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2936 class class class wbr 5030 {copab 5092 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 |
This theorem is referenced by: nfrn 5788 nfpred 6121 nffun 6347 nff1 6547 nfsup 8899 nfinf 8930 gsumcom2 19088 ptbasfi 22186 mbfposr 24256 itg1climres 24318 funcnvmpt 30430 nfwsuc 33218 aomclem8 40005 rfcnpre1 41648 rfcnpre2 41660 smfpimcc 43439 |
Copyright terms: Public domain | W3C validator |